In my case it was just a letter from the 501(c)(3) organization acknowledging the donation. I can't speak to how the valuation was done.

Goodwill or whoever has their own guidelines for how much they value stuff, presumably somewhat related to what they can sell it for.

Goodwill's system is that they give you pre-signed receipt and you fill in whatever you like.

That's not how it's supposed to work, but I won't claim that some of them don't do that.